Is there a way to drive "Mute a node" or Toggle "Auto Refresh" in the Image Texture Node in Movie / Sequence Image Mode
Those properties are not animatable. You can't add a driver to them. Although they can be used to drive other properties, as you noticed.
However as stated in the comments you can use a Mix RGB node and drive the factor field, to be 0 or 1 for True or False.

Use the first Fac to be driven by the Auto Refresh custom property, and the second Fac to be driven by the Mute property. I set it to black but you can also use a Mix shader with a transparent shader like so depending on what you want to achieve.